In a row of students, Deepak is seventh from the left and Madhu is twelfth from the right. If they interchange their positions, Deepak becomes twenty-second from the left. How many students are there in the row?

Options

A. 19
B. 31
C. 33
D. can't be found

Explanation

To solve the problem step by step, let's denote the following:

- Let \( L \) be the position of Deepak from the left.
- Let \( R \) be the position of Madhu from the right.
- Let \( N \) be the total number of students in the row.

### Step 1: Identify the positions of Deepak and Madhu
From the question:
- Deepak is 7th from the left, so \( L = 7 \).
- Madhu is 12th from the right, so \( R = 12 \).

### Step 2: Determine Deepak's new position after the interchange
After Deepak and Madhu interchange their positions, Deepak becomes 22nd from the left.

### Step 3: Set up the equation for total students
When Deepak is 22nd from the left, we can express the total number of students \( N \) using the positions of Deepak and Madhu:
- The position of Madhu from the left can be calculated as follows:
- Since Deepak is originally 7th from the left and becomes 22nd from the left after the interchange, we can find Madhu's original position from the left:

\[ \text{Position of Madhu from the left} = N - R + 1 \]

### Step 4: Substitute known values into the equation
Substituting \( R = 12 \) into the equation gives us:
\[ \text{Position of Madhu from the left} = N - 12 + 1 = N - 11 \]

### Step 5: Set up the equation based on Deepak's new position
Since after the interchange, Deepak is now at position 22 from the left:
\[ N - 11 = 22 \]

### Step 6: Solve for \( N \)
Rearranging the equation gives:
\[ N = 22 + 11 = 33 \]

### Conclusion
Thus, the total number of students in the row is \( N = 33 \).